Why is shear stress highest at the wall of a vessel?

[tajmann]

Hey guys,

I am trying to conceptualize as to why shear stress in a vessel is highest at the wall of the vessel and why it is at a minimum at the center.

First let me see if I actually understand shear stress - In a vessel with blood flow, it is the force required to overcome the viscosity that causes blood to flow at different rates.

Now, velocity of the blood (fluid) is highest at the center and lowest (~0) at the wall. I just don't understand why there is a difference b/w the center and the periphery. The way I understood it, there was a more or less gradual decrease in the velocity of blood as it reached the wall. As such, the shear stress would be more or less be equal at the wall and the center. I know I'm not understanding something. Thanks for the help. Much appreciated.

 

[Chestermiller]

The shear stress is equal to the viscosity times the derivative of the velocity with respect to radial position. The derivative of the velocity is equal to zero at the center of the tube, and highest at the surface. We know that the derivative of velocity is zero at the center of the tube, because this is the location at which the velocity is maximum.

 

[haruspex]

The shear stress is equal to the viscosity times the derivative of the velocity with respect to radial position. The derivative of the velocity is equal to zero at the center of the tube, and highest at the surface. We know that the derivative of velocity is zero at the center of the tube, because this is the location at which the velocity is maximum.

I don't think that really answers the question. Why could the velocity profile not look like 1-r/R? Msybe it's not differentiable at the centre.
Tajmann, try reading http://www.cs.cdu.edu.au/homepages/jmitroy/eng243/sect09.pdf

 

[Chestermiller]

I don't think that really answers the question. Why could the velocity profile not look like 1-r/R? Msybe it's not differentiable at the centre.
Tajmann, try reading http://www.cs.cdu.edu.au/homepages/jmitroy/eng243/sect09.pdf

The solution to the fluid mechanics equations goes as (1- (r/R)²), not 1 - r/R. Are you asking how they get the solution to the fluid mechanics equations?

 

[haruspex]

The solution to the fluid mechanics equations goes as (1- (r/R)²), not 1 - r/R.

Yes, I know. I was just pointing out that the argument you gave was not sufficient to resolve the question posed.

 

[Chestermiller]

Yes, I know. I was just pointing out that the argument you gave was not sufficient to resolve the question posed.

Is it sufficient now?

 

[haruspex]

Is it sufficient now?

A proof that it's 1-(r/R)² would suffice.

 

[Chestermiller]

A proof that it's 1-(r/R)² would suffice.

I don't feel like providing that here. The detailed derivation of the equations for laminar flow in a tube can be found in any book on transport phenomena. I suggest Transport Phenomena by Bird, Stewart, and Lightfoot.